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before the default version (which has a bug).
7.14 Isoperimetric problem. We consider the problem of choosing a curve in a twodimensional plane
that encloses as much area as possible between itself and the xaxis, subject to constraints. For
simplicity we will consider only curves of the form
C = {(x, y )  y = f (x)},
where f : [0, a] → R. This assumes that for each xvalue, there can only be a single y value, which
need not be the case for general curves. We require that at the end points (which are given), the
69 curve returns to the xaxis, so f (0) = 0, and f (a) = 0. In addition, the length of the curve cannot
exceed a budget L, so we must have
a
0 1 + f ′ (x)2 dx ≤ L. The objective is the area enclosed, which is given by
a f (x) dx.
0 To pose this as a ﬁnite dimensional optimization problem, we discretize over the xvalues. Specifically, we take xi = h(i − 1), i = 1, . . . , N + 1, where h = a/N is the discretization step size, and
we let yi = f (xi ). Thus our objective becomes
N yi , h
i=1 and our constra...
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 Fall '13
 F.Borrelli
 The Aeneid

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